Most students are introduced to the “golden mean” (approximated by the decimal number 1.618033988 and denoted as the Greek letter phi (Φ)”) as what the ancient Greeks considered to be the “perfect ratio” with the implicit suggestion if you don’t think that rectangles with a Φ:1 aspect ratio are the best damn rectangles you’ve ever seen in your life, there is
something wrong with you. Now, these golden aspect ratios make for perfectly respectable rectangles, but it was never obvious to me what made them that special. I thought it is a little odd that humanity would have a consensus on such a subjective notion as beauty. A google survey finds that this idea is alive and well in children’seducation (far more so than the tongue map from we looked at in an earlier comic). Perhaps this notion deserves some closer examination.

The number Φ can be defined in a few ways, one being the limiting ratio of successive terms in the fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34…). It has inspired generations of Φ-o-philes, the first of whom I can find seems to be Adolf
Zeising, who wrote the following in 1854 (from the wikipedia link):

“…the universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.”

We can examine a crystal structure of DNA and find that the ratio of the helix diameter to length of one turn is about 34 Å to 20 Å (1.7), that is not Φ. A universe in which the helix length to diameter ratio was Φ (or even, say, 1.62) would likely have bizzaro physics compared to ours. Nautilus shells have log-spiral coefficients of about 1.33 (not even close to Φ). Given the variance in human anatomy, there is no reason to believe that various anatomical ratios approach Φ in any meaningful way. People superimpose all sorts of thick lined golden rectangles over Parthenon pictures and blueprints, but not only are the lines positioned rather arbitrarily, there is no evidence that the builders (who worked a century before Euclid was born) were even aware of Φ. You can find hucksters on the internet trying to sell you a slice of Φ for all sorts of things including guitars , stock market trading strategies and beauty products.

What drives people to claim structures fit the golden ratio for any number in the neighborhood of 1.6 and do so in flat denial of clear facts? Perhaps it is merely the old trick of imbuing a work with authority via complicated diagrams and mathematical symbols (“Geometry! Greek letters! Math formulas! See?”). After all, we do see numbers like π (“pi” , defined as the ratio of a circle’s circumference to its diameter, approximately 3.14159…) in all sorts of important formulae . It isn’t so much that we find π in a lot of places, but more that we find circles and spheres a useful way to model physical things. Tree trunk cross sections are certainly not perfectly circular, but if we pretend they are (along with some other information), we can easily make a reasonable guess about how much a wood might be in a forest.

This isn’t to say that Φ is never involved in parameterizing models of the physical world (it is, after all, the solution to a simple quadratic equation and is easily constructed geometrically), but its use isn’t any more significant than other dimensionless numbers like the square root of two (1.4121…) or Euler’s number (2.7182…). This doesn’t mean that all scientific publications that invoke Φ are valid though, some of them are considered to be pretty stupid .

An example where there does seem to be something interesting happening is sunflower seed patterns, a favorite hypothesis of Alan Turing (the father of modern computer science). Apparently the first time anyone has ever decided to look at this hypothesis critically is in a recent project , asking students to grow and send in sunflowers so that their spirals can be counted (look at the site to see how this is done). I’m a little concerned by their sampling method bias as well as the fact that they haven’t actually released their dataset yet, but it does seem like about 80% of sunflowers do have spiral counts of fibonacci numbers (I presume the spiral counts have to be 21, 34 or 55). Perhaps there is something special going on here, perhaps there isn’t; the fact is we still don’t have a good understanding of the bio-molecular process of phyllotaxis and there could be completely ordinary reasons for why we observe these patterns in sunflowers. Recall that there are many plants that exhibit growth patterns that don’t seem to have anything to do with golden mean, so why does this allegedly universal law only show up in things like sunflowers and pinecones and not seaweed, grape bunches or maple leaves?

The most common references to Φ are the supposedly “hidden” references in the compositions of classical paintings. You may not be aware, but there is an exciting classical realist painting movement that is gaining momentum across the world today; there are more ateliers that offer instruction than ever before. Many great artists pumping out stunning and inspiring work , of which I am a big fan. Perhaps this is why I’m especially bothered when I find skilled and otherwise knowledgable people parrot the same old fibs about Φ.

A well known book amongst artists is Harold Speed’s “The Practice Of Science And Drawing” written in 1913. The very last pages in the print that I own are devoted to pointing out particular length ratios between conspicuously chosen points in classical paintings. No doubt the particular points he’s chosen lead to Φ-like ratios, but can we really believe if we selected the same sort of points in those paintings that we wouldn’t have found other ratios just as frequently? Couldn’t those ratios be just as close to say, π/2 or 5/π? The widely respected book “Classical Drawing Atelier” spends some time trying to explain the origin and importance of the golden mean while committing same factual errors that have already been cited here. The golden mean is invariably mentioned in (terrible) “how to” websites about drawing and painting, which no one should ever use for anything (especially learning how to draw and paint).

Many of us lament the poor quality of childhood art “education”, as despite taking art classes every year, few of us learned to draw past an elementary school level. We then discovered as adults that basic artistic skill can be acquired
by just about anyone with functional vision and motor control. Studying classical realism has enabled many of us to “touch the magic” by emphasizing objectivity, accuracy and critical thought. The process of painting and drawing can, at times, seem like little else other than obsessive self-interrogation: “do these two points on the figure really line up vertically?”, “is the figure outline I’ve drawn too thin or too wide?”, “does that shadow shape on my page really look like the one in I see on my subject?”. Why would anyone compromise this aspiration to integrity with mysticism and trivial lies? At its best, art is about truth; and the truth behind the intricacy and complexity of human vision is far more beautiful than any Φ fiction I can come up with. The promotion of mathematical and scientific illiteracy is only useful to those whose aim it is to obscure knowledge.

Discussion (4)

Thank you so much for sharing such an insightful feature. I have been on a campaign to debunk the lore of the Golden Mean in fine art for some time now. A fun and informative video on the subject comes from mathematician Keith Devlin (Math Encounters — Fibonacci & the Golden Ratio Exposed.) It is a somewhat show presentation and lecture (I would recommend skipping the introductory speaker as he is quite terrible. LOL! – https://www.youtube.com/watch?v=JuGT1aZkPQ0) Again—thank you!